Aptitude - Problems on H.C.F and L.C.M - Discussion

Well, I will break it into English but no maths.

Generally all the bells rang (toll) at its first encounter, that means all 6 bells started its journey by joining all their hands at once, i.e., at 0th sec<< remember this started journey>>.

Then they lost their hands and joined again at 120th second, in the mean-while they are tolling individually and not combinedly.

1. It is nothing but at 14th 19th. 25th 38th 57th 69th. Until 119th sec they didn't rang combinedly then at *120th* sec they combined their hands and rung. This is the least number that those 6 bells have waited for their combined toll (nothing but LCM in maths criteria), shortcut used.

2. Then they again rung combinedly after *120*+120th=240th sec+. 120+120+..until 1800.

3. They had given us 30min =1800 sec) , so if we divide that 1800 sec by 120sec (intervals) 1800/120=15 we will get.

Actually this (15) should be the answer if all the bells did not start combinedly, but our bad luck, they mentioned in the question that ""Six bells commence tolling together. "" that means they have started its journey itself by ringing combinedly.

4. The extra +1 is nothing but this started journey.

Now go through what @Saravan and all others said you will get more detailed idea.

Hope this understands. Thank you.

Commense is nothing but start tolling. here six bells are there and toll together at intervals 2,4,6,8,10,12

first bell tolls at every :2secs,4secs,6sec......120sec....

second bell tolls at every :4secs,8secs,12secs.....120secs.....

third bell tolls together at every:6sec,12sec,18sec..120sec...

fourth bell tolls together at every:8sec,16sec,24sec...120sec..

fifth bell tolls together at every:10sec,20sec,30sec ,....120sec...

sixth ell tolls together at every:12sec,24sec,36sec....120sec...

NOW OBSERVE ALL TOLLS the common sec in *All* bell tolls is 120sec

thats why we need to find l.c.m in shortcut
come to question:
1 toll altogether takes time=120sec
? tolls altogether takes time=1800sec(30min*60sec)

The number of tolls are 1800*1/120 = 15 tolls(times)


But in question:
Six bells commence tolling together(first toll) and also toll at intervals of 2, 4, 6, 8 10 and 12 beginning itself started from tollinf so we should consider that toll also.

Hence answer is 15 Tolls + First Toll == 16 dat's it.

Hope you understand.

I got the easiest way to understand the solution that how the answer is 16.

Look at your clock it starts from 12:00 and ends at 12:00.
Now consider each hour of your clock as a minute that means 12 to 1=1 min and 1 to 2=2min. So for 2min it's 12 to 2 = 2min.

Now starting from 12:00 start counting for 2 minutes interval taking 12:00 as 1(times the ring bell together).

So, 12:00 - 1
12:00 to 2:00 - 2 (2nd time they rang together)
2:00 to 4:00 - 3
.
.
.
.
.
10:00 to 12:00 - 7 and now you have completed your 12 minutes out of 30 minutes.

Repeat the same process you will complete 24 minutes out of your 30 minutes and 13 times the bell rang together.

Now you are left with 6 minutes.
so 12:00 - 2:00 = 14 times.
2:00 - 4:00 = 15 times.
4:00 - 6:00 = 16 times.

Finally, you have completed your 30 minutes and the bell rang 16 times.

Hope now you people could understand how the answer is 16.

Actually got solution.

All bells starting from 1 second at a time 1 time bells tolling at a time.

1st bell tolling every 2 sec means 2, 4, 6, 8,...120...240... 360....480

2nd bell tolling every 4 sec means 4, 8, 12, 16...120...240...360...480

3rd bell tolling every 6 sec means 6, 12, 18, 24...120...240...360...480

4th bell tolling every 8 sec means 8, 16, 24...120...240...360...480

5th bell tolling every 10 sec means 10, 20, 30...120...240...360...480

6th bell tolling every 6 sec means 12, 24, 36...120...240...360...480

So 1 sec =1 time tolling at a time.

120 sec = 2 time tolling at a time.

240 sec = 3 time tolling at a time.

360 sec = 4 time tolling at a time.

4800 sec = 5 time tolling.

600 sec = 6 time tolling at a time.
.
.
Last 1800 sec = 16 time tolling at a time.

Total time bells tolling at a time =16 times.

@Sathyaraj.

In this case, for every 120 seconds, all six bells ring together.
Let us look in another perspective.
At 120th second,
The first bell would have ringed for 60 times.
The second bell would have ringed for 30 times.
The third bell would have ringed for 20 times.
The fourth bell would have ringed for 15 times.
The fifth bell would have ringed for 12 times.
The sixth bell would have ringed for 10 times.

Here, the point is that even though these bells are ringed several times only at the 120th second all of these six bells ring simultaneously.
Only for this purpose the LCM is used in this problem.

A least common multiple of (2, 4, 6, 8, 10 and 12)

we can write 2 = 1 * 2

4 = 22

6 = 2 * 3

8 = 23

10 = 2* 5

12 = 22 *3.

A least common multiple of (2, 4, 6, 8, 10 and 12) = 23 * 3 * 5.

= 8 * 3 * 5
= 120 sec

Bells ring together after every 120 sec.

Required number of times in 30 minutes (30 *60 seconds ) = [(30 *60)/120]

= 15

But we have to add 1 because at starting all bells will be rung once a time after that the ring 15 times.

= 15 + 1.
=16 times.

The required number of times when they toll together is 16.

Its simple, according to the shortcut formula everyone understood why the tolling together takes place at every 120sec(2min) period from the l.c.m concept.

But why 1 is added means, it is clearly mentioned in the question that the 6 bells starts tolling together at the starting stage(first tolling is done by together) and tolls at the intervals 2, 4, 6 ,8, 10, 12. so, before tolling together at 120sec(2min), they already tolls together at the 1st(zeroeth sec i.e starting) itself. so that 1 is added.

If two bells toll after every 3 secs and 4 secs respectively and if they commence tolling at the same time.

Then the first bell tolls after every 3, 6, 9, 12 secs...
The second bell tolls after every 4, 8, 12, ....
So they toll together again after 12 secs, which is the LCM

Henceforth they toll after every 12 seconds, i.e whenever the time is a common multiple of both 3 and 4.

Since the bells start tolling together, the first toll also needs to be counted. Therefore we need to add 1.

It's asked in the question that Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?

Here as per the question the 1st bell can toll 15 times in 30 min i.e (30/2), 2nd bell can toll 7.5 times in 30 min i.e 30/4, 3rd bell 5 times, 4th bell 3.7 times, 5th bell 3 times and 6th one 2.5 times in 30 min...together they tolled 36.7 times in 30 min..

Why this approach is wrong?

Friends, Let bells ring together once in 120 seconds.
Let's say they all ring for the first time at 1st sec and they ring altogether once again after 120 sec, say 121st sec.

Therefore they ring at.

1st sec, 121st sec, 241st sec, and this series forms an ap with.

A = 1, d = 120.

Therefore.
Number of terms before 1800 =1 6 (should be) as per answer given.

But the number of terms comes as 15 and the bells ring together at 1801st sec for the 16th time.